But while some are giving up their dream, others are just arriving, further stretching resources at makeshift camps.
"CBP Commissioner Kevin McAleenan told reporters earlier Monday that as many as 1,000 people that were part of a migrant caravan traveling through Central America and Mexico sought to enter the United States illegally".
"I came here for one reason, and that's because there is a lot of violence in Honduras", said Meza, as her children played with empty tear gas canisters shot by the Border Patrol.
Ebrard told reporters Tuesday a key administration goal is securing a US commitment to development projects in Honduras, where the vast majority of the migrants in the caravan come from, as well as neighboring Guatemala, El Salvador and elsewhere in Central America. These included 29 cases in which tear gas was used and 43 incidents of pepper spray.
"I didn't know Robert Mueller needed protection, ' McCarthy told reporters, adding that there's 'nothing out there" for Mueller or Democrats to fear from Trump or his administration.
He also said the tear gas was "very safe" and the people suffering from its effects "to a certain extent were the people that were putting it out there".
U.S. Army soldiers install a barbed wire fence along Anzalduas International Bridge near the U.S. - Mexico border in McAllen, Tex., Nov. 5, 2018.
US President Donald Trump speaks during an inspection of border wall prototypes on March 13 in San Diego.
"We are asking that they be relocated", Rodriguez said, noting that some migrants had approached the school grounds to ask children for money and use the school's bathrooms.
"The agents responded with the least amount of force that we possibly could", said Rodney Scott, chief of the US Customs and Border Protection's San Diego Sector.
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"They learned their lesson", an employee of Mexico's National Migration Institute told AFP at the shelter, speaking on condition of anonymity.
Closing the border crossings "prevented a unsafe situation from getting worse", McAleenan said.
"In a border context, when you are dealing with assaultive behavior, you're addressing it to resolve the situation", he said.
The use of tear gas during Sunday's events have also sparked debate.
A Honduran migrant converses with U.S border agents on the other side of razor wire after they fired tear gas at migrants pressuring to cross into the US from Tijuana, Mexico, Nov. 25, 2018.
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Videgaray said the migrants were welcome to stay in Mexico while they waited to file asylum claims in the United States.
"Entering our country illegally, particularly over our walls is not only risky, but also very foolish", San Diego Sector Chief Patrol Agent Rodney Scott said in a statement.
The bedraggled men, women and children of a caravan of mostly Hondurans began cramming into the complex in Tijuana about three weeks ago.
"If they close the border I ask God that here in Tijuana, or in another country they open doors to us, to allow me to survive with my children", said Meza.
Department of Homeland Security spokesman Tyler Houlton said Tuesday that CBP had arrested 42 people Sunday for illegally crossing the border, revising a higher figure the agency had previously released due to what Houlton said was confusion "in the heat of the moment".
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"People here are pretty dug in", said Rubio, a Republican. A handful from the group threw rocks over a border fence at Border Patrol agents, who fired CS gas at the group.